Multiobjective Optimization Athens 2005 Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa

Contents Introduction Multiobjective Optimization MO Non-Heuristic Linear Nonlinear Handling Constraints Techniques EMOO Using Constraints in EMOO Conclusions References Introduction Multiobjective Optimization MO Non-Heuristic Linear Nonlinear Handling Constraints Techniques EMOO Using Constraints in EMOO Conclusions References

Introduction Optimization Problem Find a solution in the feasible region which has the minimum (or maximum) value of the objective function Possibilities Unique objective (function) Multiobjective multiple optimal solutions selection by preference function Solving: Non-Heuristic (deterministic) Heuristic Optimization Problem Find a solution in the feasible region which has the minimum (or maximum) value of the objective function Possibilities Unique objective (function) Multiobjective multiple optimal solutions selection by preference function Solving: Non-Heuristic (deterministic) Heuristic

Multiobjective Optimization Find a solution that: Minimize (objectives) f(x) = (f1(x), f2(x),..., fn(x)) Subject to (constraints) g(x) = (g1(x), g2(x),...,gm(x)) ≤ 0 x = (x1, …, xn) f, g linear/nonlinear functions Find a solution that: Minimize (objectives) f(x) = (f1(x), f2(x),..., fn(x)) Subject to (constraints) g(x) = (g1(x), g2(x),...,gm(x)) ≤ 0 x = (x1, …, xn) f, g linear/nonlinear functions

Multiobjective Optimization Δ (The searching space): set of all possible solutions of x. Ð ( The feasible space): set of all solutions that satisfy all the constraints. Δ (The searching space): set of all possible solutions of x. Ð ( The feasible space): set of all solutions that satisfy all the constraints.

Multiobjective Optimization Pareto Optimal x ℮ Ð is said to be Pareto Optimal if there does not exist another solution x’ ℮ Ð that fi(x) = fi(x’) i = 1, …, m fi(x) < fi(x’) i = 1, …, m at least one i. x solution dominate x’ solution Pareto Front The maximal set of non-dominated feasible solutions. Pareto Optimal x ℮ Ð is said to be Pareto Optimal if there does not exist another solution x’ ℮ Ð that fi(x) = fi(x’) i = 1, …, m fi(x) < fi(x’) i = 1, …, m at least one i. x solution dominate x’ solution Pareto Front The maximal set of non-dominated feasible solutions.

MO Non-Heuristic Multiobjective Optimization Linear (f and g) Multiobjective Simplex Method Techniques: Multiparametric Decomposition (weights a priori) Fractional Program (ratio objectives) Goals Program (goal deviations) Nonlinear (f or g) Compromise Programming Ideal Solution for each objective function Distance between solutions Objective weights Compromised Solution Compensation of objectives Competitive Objectives Multiobjective Optimization Linear (f and g) Multiobjective Simplex Method Techniques: Multiparametric Decomposition (weights a priori) Fractional Program (ratio objectives) Goals Program (goal deviations) Nonlinear (f or g) Compromise Programming Ideal Solution for each objective function Distance between solutions Objective weights Compromised Solution Compensation of objectives Competitive Objectives

MO Non-Heuristic Actual Basic Variables Basic VariablesNonbasic Variables Values of Basic Variables X1...XmX1...Xm 1 … 0. 0 … 1 Y 1 (m+1) … Y 1p. Y m (m+1) … Y mp X10...Xm0X10...Xm0 Objetives 0 … 0. 0 … 0 Z 1 (m+1) … Z 1p. Z n (m+1) … Z np f 1 (x 0 ). f n (x 0 )

Constraints-Handling Techniques Penalty Function Very easy Depends on problem  Problems with strong constraints  Repair Heuristic Useful when it’s difficult to find feasible problems Depends on problem  Separation between objectives & constraints No depends on problem Extend to multiobjective optimization problems Hybrid Methods Use of numerical optimization problem Excessive computational cost  Others Penalty Function Very easy Depends on problem  Problems with strong constraints  Repair Heuristic Useful when it’s difficult to find feasible problems Depends on problem  Separation between objectives & constraints No depends on problem Extend to multiobjective optimization problems Hybrid Methods Use of numerical optimization problem Excessive computational cost  Others

EMOO Evolutionary MultiObjective Optimization Techniques: A priori Preferences before executing Reduce the problem to a unique objective Unique solution A posteriori Preferences after executing Multiple solutions Methods: Non-based on Pareto Optimal concept based on Pareto Optimal concept Non-elitist elitist Evolutionary MultiObjective Optimization Techniques: A priori Preferences before executing Reduce the problem to a unique objective Unique solution A posteriori Preferences after executing Multiple solutions Methods: Non-based on Pareto Optimal concept based on Pareto Optimal concept Non-elitist elitist

EMOO A posteriori: Non-based on Pareto Optimal concept VEGA algorithm (Vector Evaluated Genetic Algorithm) k objectives, population size N Subpopulations size N/k Calculate fitness function and select t best individuals (create new subpopulation) Shuffle all subpopulations Apply GA operators and create new populations of size N Speciation Problem: select individuals depending on 1 objective only A posteriori: Non-based on Pareto Optimal concept VEGA algorithm (Vector Evaluated Genetic Algorithm) k objectives, population size N Subpopulations size N/k Calculate fitness function and select t best individuals (create new subpopulation) Shuffle all subpopulations Apply GA operators and create new populations of size N Speciation Problem: select individuals depending on 1 objective only

EMOO A posteriori: non-elitist based on Pareto Optimal concept MOGA algorithm (MultiObjective Genetic Algorithm) range (x) = 1 + p(x) (p(x) number of individuals that dominate it) Sorting by minimal range Create a dummy fitness (lineal or non-linear) and calculate (interpolate) depending on individual range Select t best individuals (niches) Apply GA operators and create new population Others: NSGA, NPGA, … A posteriori: non-elitist based on Pareto Optimal concept MOGA algorithm (MultiObjective Genetic Algorithm) range (x) = 1 + p(x) (p(x) number of individuals that dominate it) Sorting by minimal range Create a dummy fitness (lineal or non-linear) and calculate (interpolate) depending on individual range Select t best individuals (niches) Apply GA operators and create new population Others: NSGA, NPGA, …

EMOO A posteriori: elitist based on Pareto Optimal concept NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm) Population P (size N) Create new population P’ (size N) using GA operators Merge both populations and create new Population R (size 2N) Sort by range of domination Select t individuals (tournament & niches) and create a new population R’ Others: DPGA, PESA, PAES, MOMGA, … A posteriori: elitist based on Pareto Optimal concept NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm) Population P (size N) Create new population P’ (size N) using GA operators Merge both populations and create new Population R (size 2N) Sort by range of domination Select t individuals (tournament & niches) and create a new population R’ Others: DPGA, PESA, PAES, MOMGA, …

Using Constraints in EMOO Eliminate non-feasible solutions Use Penalty functions Separate solutions feasible and non- feasible Define problem with Goals Eliminate non-feasible solutions Use Penalty functions Separate solutions feasible and non- feasible Define problem with Goals

Conclusions Multiobjective Optimization Non-Heuristic Multiobjective Simplex Method (Linear) or Compromised Programming (Non-Linear) Using Constraints Goals, Penalties, Weights… Heuristic Using GA (EMOO) A priori (unique objective) A posteriori Using GA and Constraints Multiobjective Optimization Non-Heuristic Multiobjective Simplex Method (Linear) or Compromised Programming (Non-Linear) Using Constraints Goals, Penalties, Weights… Heuristic Using GA (EMOO) A priori (unique objective) A posteriori Using GA and Constraints

References Gracia Sánchez Carpena. Diseño y Evaluación de Algoritmos Evolutivos Multiobjetivo en Optimización y Modelación Difusa, PhD Thesis, Departamento de Ingeniería de la Información y las Comunicaciones, Universidad de Murcia, Murcia, Spain, November, 2002 (in Spanish). Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, March 2002, ISBN David A. Van Veldhuizen. Multiobjective Evolutionary algorithms: Classifications, Analyses, and New Innovations. PhD thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, May J. David Schaffer. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, Tadahiko Murata. Genetic Algorithms for Multi-Objective Optimization. PhD thesis, Osaka Prefecture University, Japan, Kalyanmoy Deb, Associate Member, IEEE, Amrit Pratap, Sameer Agarwal, and T. Meyarivan A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-II Kaisa Miettinen Nonlinear Multiobjective Optimization Kluwer Academic Publishers, Boston, 1999 Gracia Sánchez Carpena. Diseño y Evaluación de Algoritmos Evolutivos Multiobjetivo en Optimización y Modelación Difusa, PhD Thesis, Departamento de Ingeniería de la Información y las Comunicaciones, Universidad de Murcia, Murcia, Spain, November, 2002 (in Spanish). Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, March 2002, ISBN David A. Van Veldhuizen. Multiobjective Evolutionary algorithms: Classifications, Analyses, and New Innovations. PhD thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, May J. David Schaffer. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, Tadahiko Murata. Genetic Algorithms for Multi-Objective Optimization. PhD thesis, Osaka Prefecture University, Japan, Kalyanmoy Deb, Associate Member, IEEE, Amrit Pratap, Sameer Agarwal, and T. Meyarivan A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-II Kaisa Miettinen Nonlinear Multiobjective Optimization Kluwer Academic Publishers, Boston, 1999

Multiobjective Optimization Athens 2005 Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa Department of Architecture and Technology Universidad Politécnica de Madrid Santiago González Tortosa